### Video Transcript

Which of the graphs a, b, c and d
most correctly shows how the velocity of an object changes with time if the object
is subject to a constant force while moving through a fluid that exerts a drag force
on it, bringing it to its terminal velocity?

Okay, as we get started here, we’re
told that we have some kind of object. And let’s say that this here is our
object. We’re told further that our object
is subject to a constant force. We can sketch that in on our
object, say acting to the right. And we’ll simply call that force
𝐹. We don’t know what it is, but it’s
just some force. In addition to this, we know that
our object is moving through a fluid. A fluid recall is a liquid or a
gas. And as a result, as the object
moves, it experiences a drag force. That’s a force resisting its
motion. Lastly, we know that the effect of
this drag force is such that eventually our object is brought to its terminal
velocity. We want to know which of these four
graphs — a, b, c and d — most correctly represents how the velocity of our object
changes in time.

To begin figuring that out, let’s
clear a bit of working space on screen. Okay, so here again is our object,
and it’s being acted on by this constant force we’ve called 𝐹. We know our object is moving
through a fluid. And what if we just say that that
fluid is water, that our object is moving through a tank of water? Well, in that case, we know what
will happen as our object starts to move under the influence of the force 𝐹. As it begins to move to the right,
there will be a drag force that emerges acting in the opposite direction. We can call it 𝐹 sub 𝐷 that
opposes this motion.

And there’s an important difference
between the drag force and this force we’ve called 𝐹. Recall that 𝐹 is a constant
force. It never changes, whereas the drag
force does change as the velocity of our object changes. In particular, the drag force will
increase the faster our object moves. Now, before we get too far ahead of
ourselves. Let’s notice that in all four cases
for all four of our answer options, the initial velocity of our object is said to be
zero.

So let’s do this. Let’s let our object in this tank
of water start from rest. Beginning at that point, our object
speed will clearly be zero. And if it has no speed, then
there’s no force resisting its motion, since it has no motion. At the outset then, at the initial
moment, we can say that there is no drag force on our object. The only force acting on it is this
constant force 𝐹. Under the influence of that force,
though, because it is a net force acting on our object, our object starts to
accelerate. And the instant our object speeds
up, the moment that it has a speed above zero, a small drag force appears opposing
that motion. This drag force is caused by the
interaction between this object and the water in the tank.

So our object’s velocity started at
zero. And it’s increasing thanks to the
fact that there are unbalanced forces acting on the object. And therefore, it accelerates. But here is where we want to recall
that the drag force, unlike our constant force 𝐹, doesn’t stay the same. It’s generally the case that drag
force increases as object velocity increases. And indeed, thanks to the fact that
there’s an overall or net force on our object to the right, its velocity will
increase. But then, as its velocity
increases, so does the drag force. We see, though, that at this point
the drag force is still less than the constant force 𝐹. So our object continues to speed
up. And as it does so, the drag force
grows and grows.

So long as these forces — our drag
force and our constant force — are unbalanced, our object will continue to speed
up. But the more it does so, the
greater the drag force until eventually the two forces are equal. At this point, we can recall that
when the net force on a projectile such as our object is zero, that means that
object has reached what’s called terminal velocity. This is the point at which a
projectile is no longer speeding up, but it’s reached its maximum speed. Because an object’s terminal
velocity is its maximum speed, that means we can eliminate a couple of our possible
answer options.

We see both for graph a as well as
for graph d that the far-right portion of the graph in each case does not indicate a
maximum velocity. In the case of graph a, that’s
because we’ve already achieved a higher velocity earlier on. So graph a doesn’t correctly show
an object reaching terminal velocity. Then, in the case of graph d, our
velocity is still increasing as we get to the right most portion of the curve. In this case also, we’re not at a
maximum value. We’re not at terminal velocity. If we look then at graph c, this
graph shows us an object that is speeding up initially at an increasing rate. In other words, not only was the
object getting faster, but the rate at which it got faster was growing.

But in the case of our object,
acted on by the constant force 𝐹 and opposed by the drag force, what we saw instead
was that while the object’s velocity did increase, it was increasing at a decreasing
rate. That is, as the object’s velocity
approached its maximum value, its velocity changed by smaller and smaller
amounts. So it’s the shape of this part of
the curve of graph c, which tells us that this is not a correct representation of
our object’s velocity versus time. Yes, at first, our object was
speeding up. But the rate at which it was
speeding up was not increasing. It was decreasing. So option c can’t be our choice
either.

When we look at the graph for
option b, we see that this graph has the right shape to it. Velocity is increasing but
increasing at a decreasing or smaller rate over time. And eventually, the velocity does
level out to a set value, the object’s terminal velocity. It’s graph b then that correctly
represents the relationship we’re looking for.